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Matrices
*
*
The matrix A is shortly written as A a ij
and the matrix B is shortly written as B bij
23
32
Matrices are used in Engineering,
Economics, Statistics, Chemistry, Physics etc
...
The following are row matrices
511 , 2
7 12 ,
0
3
913
The order of any row matrix is 1 n
...
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
*
Difference between Matrix and Determinant :
i) A matrix can not have a definite value but
determinant have a definite value
...
iii) Elements of matrix are enclosed in brackets
Discovery :
A British mathematician Arthur Cayley,
formulated the general theory of “Matrices” in
1857
...
If a matrix contains m rows and n columns, we
say that it is an m × n (read m by n) matrix and
m × n is called the order of the matrix
...
The numbers which form a
matrix are called as Elements of matrix
...
g
...
4
3
711 , 2 , 0
2 x1
7 3 x 1
The order of any column matrix is m 1
...
The following are Rectangular Matrices
3
1 2 3
2 , 3 2 1 , 1 2 31 3
2 1
1 3
The following are column matrices
MHT – CET / JEE (Main)
[1]
3)
4)
Square Matrix :
If number of rows of a matrix are equal to
number of its columns (m = n), then the matrix
is called a Square Matrix
...
In square matrix A= [aij],
diagonal formed by aij, i = j is called as
Principle Diagonal
...
g
...
Diagonal Matrix :
A square matrix whose all non–diagonal
elements are zero is called a Diagonal Matrix
...
e
...
3 0 0
e
...
0 5 0
0 0 1 3 3
Number of zeros in a diagonal matrix is given
by n2 – n, where n is order of the matrix
...
i
...
if i) aij = a for i = j,
Matrices
ii) aij = 0 for i ≠ j
...
…L)’ = L’……
...
g
...
It is
also called a Null Matrix
...
i
...
if aij = 0, for i > j
3 1 5
1 2
e
...
0 2 1 ,
0 1 22
0 0 4 3 3
b) Lower Triangular Matrix :
A square matrix whose every element above the
diagonal is zero, is called as Lower Triangular
Matrix
...
e
...
4 0 0
1 0
,
e
...
3 1 0
2 3 22
6 2 5 3 3
Note: Minimum number of zeros in a triangular
n2 n
matrix is given by
, where n is order of
2
matrix
...
It is denoted by A’
or A t
...
i
...
if i) aij = 1, for i = j,
ii) aij = 0 for i ≠ j
I1 = [1] is the identity matrix of order 1
...
0 1
* (KA)’ = KA’
[2]
* (A’)’ = A
* AT A
11) Symmetric Matrix :
If the square matrix & its transpose are same
then it is called as Symmetric Matrix
...
e
...
i
...
aij = aji , i ≠ j
...
g
...
1 2
12) Skew Symmetric Matrix :
A square matrix A is said to be Skew Symmetric
if A ' A
...
i
...
if i) aij = –aji , i ≠ j &
ii) aij = 0, i = j
...
g
...
ii) A – A’ is a skew symmetric matrix
...
ii) If A and B are symmetric matrices of the same
order then
i) A + B is a symmetric matrix
...
iii) AB – BA is a skew symmetric matrix
...
A A' A A'
i
...
A
2
2
Where
A A'
A A'
is symmetric and
is
2
2
skew symmetric matrix
...
Matrices
e
...
Let
Addition or subtraction of two matrices is
nothing but addition or subtraction of their
corresponding elements
...
*
*
If A = diag d1,d 2 ,d3 ,
...
,
dn
d1 d 2 d 3
Also, A m diag d1m ,d 2 m ,d 3m ,
...
Useful Results about Determinant of Matrix :
i)
The determinant of only square matrix
exists
...
iii) The determinant of a diagonal matrix is the
iv) The determinant of a triangulate matrix is
v) If A is a square matrix and n N then
| An | = | A |n
14) Singular Matrix :
A square matrix A is said to be Singular if |A|=
0
...
A square matrix A is said to be Non–Singular if
| A | ≠ 0
...
16) Involutory matrix :
A square matrix A is said to be Involutory
matrix if A2 = I
...
i
...
a11 = b11, a12 = b12
...
Addition & Subtraction of Matrices
If two matrices are of the same order, then only
they can be added or subtracted
...
MHT – CET / JEE (Main)
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
*
Note :
If A is orthogonal matrix then
| A |= ±1 and A–1 =AT
If A and B are two orthogonal matrices, then
AB and BA are both orthogonal matrices
...
i
...
A + B = B + A
ii) Associative law
Addition of matrices is associative
i
...
(A + B) + C = A + ( B + C)
iii) Existence of additive Identity
A+0=0+A=A
Provided that matrices are conformable for
iv) Existence of additive inverse
A + (–A) = A – A = 0
Here (–A) is called as Additive Inverse of
A
...
Then the matrix whose
product
diagonal
elements
...
elements of the matrix A is called the negative
of the matrix A and is denoted by –A
...
Thus if A = [aij]m × n then kA=[kaij]m × n
Properties of scalar multiplication :
Let A=(aij)m × n and B= (bij)m × n be any matrices,
m & n are scalars then
i) (m + n) A = mA + nA …
...
i
...
[A]m × n × [ B ]n × p = [A B]m × p
Laws of Matrix Multiplication :
i) Associative law
[3]
Matrices
i
...
AB ≠ BA
...
If A is the given
matrix then A–1 exists if A is non singular
matrix
i
...
| A | ≠ 0, i
...
A is invertible matrix
...
i)
Elementary Row Transformations :
To find the inverse of given matrix by using this
method use AA–1 = I
...
Note : *(AB)–1 = B–1 A–1
...
A
* A ' (A 1 )', (A 1 ) 1 A ,
1
iii) Method of adjoint of matrix :
a b c
If A = d e f and | A | ≠ 0
g h i
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
A(BC) = (AB)C
ii) Distributive laws
A(B + C) = AB + AC …left distributive law
(A + B)C = AC + BC… right distributive law
iii) (kA)B = A(kB) = k(AB), k is any scalar
...
v) Multiplication by Zero Matrix
...
vi) Matrix multiplication is not commutative
...
It is denoted by adj
...
a b
d b
e
...
If A =
then Adj
...
A | = 0
...
The determinant
obtained by removing it’s ith row and jth column
in | A | is called as Minor of element aij
...
3 5 1
If A = 1 2 0 then
2 3 4
minor of (–3) =
minor of (–2) =
[4]
3 1
1
0
3 1
2
4
=1
= 14
*
Cofactors :
Let A be the square matrix
...
It
is denoted by cij, Aij
In above example,
∴ Cij = Aij = (–1)i+j ×
mij
Cofactor of –3 = (–1)3+2 × 1
= –1
2+2
Cofactor of –2 = (–1) × 14 = 14
*
Solution of Linear Equations :
We can solve the given linear equations by
using
i) Reduction method ii) Method of inversion
...
A
|A|
–1
MHT – CET / JEE (Main)
*
Some Useful Results :
If AB = AC & | A | ≠ 0 then B = C
AB = 0, does not mean that A = 0 or B = 0
...
(A – B)2 = A2 – AB – BA + B2
...
If A is a square matrix of order n and k is scalar
then | KA | = Kn | A |
If A and B are non singular matrices of same
order then adj(AB) = (adj
...
A)
Matrices
*
*
adj A 1 adj
...
A A
n 1
ii) adj(kA) k n 1adj(A)
iii) adj
...
A
n 12
iv) adj
...
A
iv) adj A
n
n
, where n is order of A
, n N
1
1
1
vi) A 1
A
adjA
A
A
1
1
vii) adjA
viii)
A
A
1
KA 1 A1, K 0
K
cos sin
ix) If A
, then
sin cos
cos n sin n
An
...
The adjoint of a diagonal matrix is a diagonal
matrix
...
If A is symmetric, then A–1 is symmetric
...
Inverse of symmetric matrix is symmetric
...
All the entries in the inverse of a matrix A are
v) A 1
*
*
*
*
*
*
*
*
integers if and only if A 1
...
Value of determinant of matrix A is obtained by
sum of product of elements of a row or a
column with corresponding cofactors
...
: a11c 21 a12c22 a13c23 0
A (adj
...
A) A= | A | I, where A is
square matrix and I is identity matrix of order n
...
1 5 3
e
...
If A= 2 3 1 then
2 6 5
trace of A is 1 – 3 + 5 i
...
3
Note :
i)
ii)
iii)
iv)
Trace(A) = Trace(A’)
Trace(A + B) = Trace A + Trace B
Trace(KA) = K Trace A
...
Trace B
*
Cayley Hamilton Theorem :
Every square matrix satisfies its characteristics
equation
...
*
Homogeneous and Non Homogeneous System
of Linear Equations :
i) Homogeneous System :
If AX = B and B = 0 i
...
AX = 0
e
...
2x + 5y = 0, 3x – 2y = 0 is a
homogeneous system of linear equations
...
g
...
*
Solution of Non Homogeneous System of
Linear Equations :
a) If AX = B, B 0 and A is non singular
i
...
A 0 then system has unique solution
i) Matrix Method
ii) Reduction Method
b) If AX = B, B 0 and A is singular
i
...
A 0 then system may be consistent
with infinitely many solutions or system
may be inconsistent
...
ii) If A 0 , then the system has infinitely
6)
MULTIPLE CHOICE QUESTION
1)
The values of x for which the matrix
2
x x
2
x x will be non-singular are
x 2 x
2)
a) 2 x 2
b) For all x other than 2 and -2
c) x 2
d) x 2
1 2 3
In order that the matrix 4 5 6 be non 3 5
singular , should not be equal to
a) 1
3)
If a ij
b) 2
c) 3
d) 4
1
3i 2 j and A a ij 22 , then A 1 is
2
equal to
4)
1 / 2 2
a)
1/ 2 1
1 / 2 1/ 2
b)
1
2
2
2
c)
1 / 2 1/ 2
2 / 3 1/ 3
d)
4 / 3 1
If A is square matrix for which a ij i 2 j2 , then
A is
a) Zero
b) Unit
c) Symmetric matrix
d) Skew Symmetric matrix
1 2 2
5) If A 2 1 2
a 2 b
is a matrix satisfying
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
many solutions
...
Then A100
1
b) 299 A
c) 298 A
d) A
1
1 x
sin x tan
1
If A
1 x
1
sin cot x
1
1 x
s in x tan
1
B
1 x
1
sin tan x
Then A – B is equal to
a) I
9)
b) 0
c) 27
d)
1
I
2
If A is a square matrix such that A 2 A ,
then I A A is equal to
3
a) A
10)
b) I – A
c) I
d) 3A
1 2 3 1 2
4 5 6
If P 2 3 4 2 0
then
0
0 1
3 4 5 0 4
p 22
a) 40
b) 40
c) 20
d) 20
AA T 9I3 ,
MHT – CET / JEE (Main)
[6]
Matrices
1 a 2
11) The matrix A 1 2 5 is not invertible if a
2 1 1
17)
has the value
a)
2
b) 1
13)
6
b)
b) 5
d)
3
2
c) 1
d) 25
If A and B are square matrices of the same
order such that A B A B A 2 B2 , then
ABA
1 2
2
a) A B
15)
c)
Suppose A is a matrix of order 3 and
B A A 1
...
A is equal to………
...
a) 1
19)
b) –1
c) 2
d) 3
0 1
2
If
, then adj
...
1
0
3 2
a)
2 3
3 2
b)
2 3
3 2
c)
2 3
3 2
d)
2 3
1 2
k 0
20) If A
and B
and the sum of
3 6
1 2
all elements of adj
...
AB
...
a) 3
MHT – CET / JEE (Main)
b) 16
b) – 3
c) 20
d) – 20
Matrices
x 2 5x 9 0 , then adj
...
kA k
n
2
31
...
A , then the sum of all
a) 9A
possible values of n is……
b) – 11
c) 11
d) – 1
24) If B is a 3 3 matrix such that B2 0 , then
determinant of adjoint of [ I B 2B ] is
2
a) 2
b) 0
c) 1
d) – 1
25) If A and B are 3 3 matrices such that A 2
and B 1 , then the determinant of adj
...
A adj
...
A A
n 1
c) adj
...
A
T
d) adj
...
A adj
...
8 0
...
A lies in the
0
...
8
interval …
...
X)
t
z
is………
y
t
a)
z x
z
t
b)
y x
t y
c)
z x
t z
d)
y x
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
a) 1
b) 729 I
c) 9 I
d) 81 I
30) If A is a square matrix of order n, such that
A 3 and adj
...
M 8
...
a) 64
b) 8
c) 2
d) 4
4
5 6
32) Let A
and B
,
3 3
4 3
then for what value of , adj
...
B ?
a) 0
b) –1
c) 1
d) –3
1 4
5 1
33) Let A
and B
, what value
3 2
3 k
must k have, so that A(adj
...
B) ?
a)
17
5
b)
17
5
c)
7
5
d)
7
5
2 3
34) If A
, then adj
...
4 1
72 84
a)
63 51
51 63
b)
84 72
51 84
c)
63 72
72 63
d)
84 51
35) The element of second row and third column in
1 2 1
inverse of the matrix 2 1 0 is ……
1 0 1
a) 1
b) –1
c) 2
d) –2
29) If a square matrix A of order 3 is a solution of
MHT – CET / JEE (Main)
[8]
Matrices
36) If both the matrices A and B are non-singular
then the value of x is equal to……
...
b) symmetric matrix
37) If A a ij
d) a skew-symmetric
i j, if i j
and a ij
,
2 2
i j, if i j
then A 1 …
...
The value of 3AB1 is………
...
A 2 2A
2 1
a)
1
25
b) 5
c) –5
d) 48
1
...
1 3 , then det
...
A
...
a) 5
d)
c) 25
b) 1
1
5
46) Square matrices L, M, N and P are of same
order and invertible such that L MN 1P
...
A 1
3 1 0
1 1 3
2
b)
1 1 3
2 0 2
d) 1
3 1
a)
21
c)
48) If A 1
1 1 3
3 1 2
1 3
2
1 5 3
and A 2 xA yI 0
...
a) (9, –14)
b) (–1, 14) c) (1, 14) d) (–9, 14)
cos sin
1 0
, B
49) If A
and
sin cos
1 1
1
is equal to…
cos sin
b)
sin cos
1 0
a)
1 1
cos sin
d)
sin cos
1 0
c)
1 1
50) If A is non-singular symmetric matrix, then
1
is…
a) a scalar matrix
...
denotes the greatest
integer
function,
then
the
det 3P 2 QR 1 is equal to …
a) 2
b) 3
value
c) 0
of
d) 4
52) If is a cube root of unity and
1 1
1
A
1
3
1 2
a) A
1
1
2 , then A 2 is
b) A 3
c) I
d) A 2
1 1 1 x 4
53) If 2 1 3 y 0 , then 2x + y + z
1 1 1 z 2
=……
a) 0
b) 4
c) 2
d) –2
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
C ABA T , then A T C1A
adj
...
1
60) If A = [1, 2, 3], B= 2 , C=[1, 3, 1] and
3
ABC = [p q r], then p, q, r are
a) 14, 42, 14
b) 42, 14, 42
c) 14, 42, 42
d) 42, 14, 14
61) Consider the system of equations in x, y, z as
x sin 3θ – y + z = 0
x cos 2θ + 4y + 3z = 0
2x + 7y + 7z = 0
If this system has a non–trivial solution, then
for any integer n, values of θ are given
( 1) n
( 1) n
a) n
b)
n
3
4
54) If A is square matrix, A’ its transpose,
MHT – CET / JEE (Main)
[ 10 ]
Matrices
( 1) n
c) n
6
d)
1/2 1/ 2
1/ 2 2
a)
b)
2 1/ 2
1/ 2 1
1 1/ 2
1/ 2 2
c)
d)
1
1/ 2
1/ 2 1
65) For equations x + y + 7z = 2, x – y + 5z =1,
9x – 6y – 9z = 1, values of x, z are…
1 1
1 1
a) ,
b) ,
c) 1, 3 d) 1, - 3
2 3
2 6
2 0 3
66) If A 4 3 1 is expressed as the sum of a
5 7 2
symmetric and skew–symmetric matrix, then the
symmetric matrix is
2 2 4
2 4 5
a) 2 3 4
b) 0 3 7
4 4 2
3 1 2
1 0 0
d) 0 1 0
0 0 1
2 2
1
67) The inverse of 1 3
0 is
0 2 1
3 2 6
a) 2 3 4
3 4 6
1 2 6
b) 1 1 2
2 2 5
3 2 6
c) 1 1 2
2 2 5
3 6 2
d) 1 2 1
2 5 2
MHT – CET / JEE (Main)
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
0 1
2 0 1
62) If A=
& B 2 3 then AB is
1 2 3
1 1
a) singular matrix
b) non – singular
c) |AB| = 4
d) scalar
...
b) 3, 3, –11
c) 3, –3, 11
d) 3, 3, 11
5
74) If for matrix A, A = I, then A–1 =
a) A
b) A2
c) A3
d) A4
[ 11 ]
Matrices
85)
4 2
75) If A=
then A 2I A 3I =
1 1
a) I
b) 2I
c) 0
d) 3I
76) If A & B are any two matrices such that AB = B
and BA = A then A 2 B2
n n
elements aij = 0, where
a) i < j
b) i > j
c) i = j
d) i ≥ j
79) If A is [A]3×4 matrix and B is a matrix such that
A' B and BA' both are defined then B is of
type
a) 4 × 3 b) 3 × 4
c) 3 × 3
d) 4 × 4
1 1 0
80) A 1 2 1 , which of the following is
2 1 0
correct
a) A3 3A2 I 0
b) A3 2A 2 I 0
c) A3 3A2 I 0
d) A3 A 2 I 0
x
81) If A
1
a) 1
1
82) If A
3
1
and A = A–1
...
b) n,n 0, 1, 2,
...
2
n
, n 0, 1, 2,
...
a) A collection of real numbers
b) An array of real numbers
c) An array of real or complex nos
...
78) In a lower triangular matrix, A= a ij
, the
If is the complex cube root of unity, then
0
0
2
inverse of 0
0 is
0 0 3
0 0
a) 0 0
0 0 2
2 0 0
b) 0 0
0 0 1
3 0 0
0 0
c) 0 0
d) 1 0 0
2
0 0 1
0
0
86) If A and B are symmetric matrices of order n,
then
a) A+B is skew symmetric
b) A+B is
symmetric
c) A + B is diagonal
d) A+B is zero
matrix
87) If A and B are skew symmetric matrix of order
n then A + B is
...
matrix
g h 0
a) diagonal
b) upper triangular
c) symmetric
d) skew–symmetric
90) If A is a square matrix of order n, then |Adj A| =
a) A
n 2
91) If A a ij
b) |A|n–1
2 2
c) |A|
d) |A–1|
, where aij = i + j then A=
...
A =
a) –1
b) 0
c) 1
d) none
103) For 2 × 2 matrices A, B & I, if A + B = I and
2A – 2B = I, then A =
MHT – CET / JEE (Main)
3/4 0
a)
3/4 0
3/4 0
c)
0 3/4
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
93) If A and B are square matrices of same order
then (A+B)2 = A2 + 2AB + B2 if
a) AB = –BA b) AB = BA c) A2=A d) B2 = B
94) If A and B are two square matrices of same
order three then
a) (AB)' =A' B'
b) AB=0 ⇒A = 0 or B = 0
c) AB = 0 ⇒|A| = 0 and |B| = 0
d) AB = 0 ⇒ |A| = 0 or |B| = 0
1 1 1
95) A 0 2 3 , B = (adj A) and C= 5A then
2 1 0
1 0
104) If A 0 1
0 0
a) 3Ab) –3A
0 1
105) If A 0 0
1 1
0 3/4
b)
3/4 0
0
3/4
d)
0 3/4
0
0 , then A2 + 2A =
1
c) 2A
d) –2A
0
1 then A3 + A =
0
a) 2
b) 3I3
c) I3
d) I2
i 0
106) If A
, n ϵ N, i 1 then A4n =
0 i
a) –I
b) 2I
c) 4I
d) I
107) If A and B are 2 square matrices such that
B = –A–1 BA then (A+B)2 =
a) A2 – B2 b) A2 + B2 c) 2A – 2B
d) A + B
108) The sum of products of elements of any row of a
det
...
A then the expression
a11 c11 a12 c12 a13 c13 a14 c14
a) 0
b) –1
c) 1
d) |A|
110) Choose correct statement
...
d) a square matrix whose each element is 1 is
an identity matrix
...
(d1, d2, d3)
b) diag
...
d1n ,d 2n , d 3n
d) none
112) For a square matrix A it is given that AA’ = I,
then A is
a) diagonal matrix
b) orthogonal matrix
c) symmetric matrix
d) none
2 0 1
113) If A = 3 1 2 then adj(A) is
1 1 2
[ 13 ]
Matrices
0 1 1
a) 8 3 7
4 2 2
then f
...
G
b) f
...
G
d) G
...
A(ϕ) =
sin cos
a) 81
b) –81
c) 27 d) –27
a) A(θ – ϕ) b)A(θϕ) c) A(θ/ϕ) d) A(θ + ϕ)
120) Let A be an invertible matrix then which of
following is not true?
a) (AB)’ = B’A’
b) (A2)–1 = (A–1)2
c) (At)–1 = (A–1)t
d) A–1 = |A|–1
cos sin 0
121) If f sin cos 0 , then f
...
matrix
a) skew symmetric b) symmetric
c) orthogonal
d) none
133) If A and B are symmetric matrices of same
order then AB – BA is
...
a) 7K2
b) 7K
c) 73 K d) 7 K3
[ 14 ]
Matrices
3 5 7
A 4 2 1
0 3 8
135) If
then trace (3A) =
...
sin
cos
cos3 sin 3
cos3
a)
sin 3
cos3
sin 3
c)
sin 3 cos3
cos 3 sin 3
b)
sin 3 cos 3
cos3
d)
sin 3
sin3
cos3
x
3
1 1 2
157) Let X = y , D = 5 and A = 2 1 1 ,
z
11
4 1 2
if X = A–1 D, then X is equal to :
8/ 3
8 / 3
8 / 3
1
a) 0
b) 1/ 3 c) 1
d) 1/ 3
0
0
1
2
1 2 3
158) If A = 1 3 4 , then |A–1| is :
3 4 3
a) 1/4
b) 4
c) –1/4
d) –4
0 3
–1
159) If A =
and A = λ(adj
...
adj
...
matrix
2
Matrices
a) diagonal
c) singular
168)
If
b) skew–symmetric
d) non –singular
1 0 2
1 1 2 and adj
...
P n 0
(0 denotes the null matrix) then P–1 is :
a) Pn
b) – Pn c) –(1 + P +
...
|adj A|
0 3
is equal to :
a) (3)3
b) (3)6
c) (3)9
d) (3)12
3 3 4
180) If A 2 3 4 then A–1 equals to
0 1 1
a) A2
b) A4
c) A
d) A3
181) Let A be any 3 × 3 invertible matrix
...
adj A
1
b) adj (adj(A)) = A
...
1 1
1 1
D A
0 Also, D x D y
0
3 3
3 3
c) adj(A) A
...
A
then,
for all i and j, the co - factor Cij of aij is such that
a) Cij = aji b) Cij = –aji c) Cij = aij d) Cij =(
aij)2
4 1
183) If A
,then the determinant of the
3 1
matrix A 2016 2A 2015 A 2014 is
a) 2014
b) –175
c) 2016
d) –25
184) Which one of the following statements is true
a) Non- singular square matrix does not have a
unique inverse
b) Determinant of a non-singular matrix is zero
c) If A' = A , then A is a square matrix
d) If, A ≠ 0 then |A
...
g
...
e
...
a1x b1 y c1z d1 ,
a 2 x b2 y c2 z d 2 ,
a 3 x b3 y c3z d3 , d1 or d2 or d3 0
is non homogeneous system in three
unknowns
...
MHT – CET / JEE (Main)
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
1/ 3 2 / 3 2 / 3
182) If A 2 / 3 1/ 3 2 / 3 a ij ,
33
2 / 3 2 / 3 1/ 3
1 1
0 i
...
A 0
2 3
iii) No Solution :
Consider equations
x + y = 1,
3x + 3y = 4
Here we cannot find values of x and y
satisfying these equations
there is no solution
...
This system is of the form AX = B, B 0
This system has
a1 b1 c1
i) Unique solution if D A a 2 b 2 c 2 0
a 3 b3 c 3
ii) Infinite Solutions if
a1
b1
D A a2
b2
a3
b3
a1
Dy a 2
a3
d1
d2
d3
c1
d1
b1
c1
c 2 0 , D x d 2 b 2 c2 0,
c3
d 3 b 3 c3
c1
a1
c2 0, D z a 2
c3
a3
b1
b2
b3
d1
d2 0
d3
ii) No Solution if
a1
b1
c1
D A a2
b2
c 2 0 and
a3
b3
c3
Matrices
b1
c1
Dx d2
b2
c2 0
d3
b3
c3
a 2 x b 2 y c 2 z 0,
or
a 3 x b 3 y c3 z 0
a1
d1
c1
a1
b1
d1
Dy a 2
d2
c 2 0 or D z a 2
b2
d2 0
a3
d3
c3
b3
d3
a3
Homogeneous System of Linear Equations :
If AX = B and B = 0 i
...
AX = 0
e
...
2x + 5y = 0, 3x – 2y = 0 is a
homogeneous system of linear equations
i
...
constant terms are zero
...
Types of Solutions :
i) Unique Solution :
Consider equations
x + y = 0,
2x + 3y = 0
Here x = 0 and y = 0 is the unique solution
...
e
...
there are infinite solutions
...
e
...
*
Trivial Solution or Zero Solution :
If values of all unknowns i
...
x, y, z in the
system are zero then system has trivial or zero
solution
...
e
...
*
Solution of a Homogeneous System of
Linear Equations :
Consider the system
a1x b1 y c1z 0,
MHT – CET / JEE (Main)
Here, d1 d 2 d3 0
This system is of the form AX = 0,
This system has
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
*
d1
[ 19 ]
i)
Unique or Trivial or Zero solution if
a1
A a2
b1
b2
c1
c2 0
a3
b3
c3
ii) Infinite or Non Trivial or Non Zero solution if
a1 b1 c1
A a 2 b2 c2 0
a 3 b3 c3
Note :
i) Consistent equations may have unique or
infinitely many solutions
...
The value of , such that the following system
of equations has no solution, if 2x – y – 2z = 2,
x + 2y + z = – 4 and x y z 4
Matrices
a) 3
b) 1
c) 0 (Zero)
7) If the system of linear equations
x1 2x 2 3x 3 6 , x1 3x 2 5x 3 9
d) – 3
c) a R 8 and b R 15
d) a = 8, b = 15
8) The set of all value of , for which the system
of linear equations 2x1 2x 2 x 3 x1 ,
2x1 3x 2 2x 3 x 2 , x1 2x 2 x 3 has a
non - trivial solution ,
a) Is an empty set
b) Is a singleton
c) Contains two elements
d) Contains more than two elements
9) The system of linear equations x y z 0 ,
x y z 0 , x y z 0 has a non trivial
5x y 3z y , 3x 5y z z has infinite
number of solutions is
a) 1
b) 2
c) 3
13) If the
system
of
linear
x 2ay az 0 , x 3by bz 0,
d) 6
equations
x 4cy cz 0 has a non zero
solution, then a, b, c
a) are in A
...
b) are in G
...
c) are in H
...
d) satisfy a + 2b + 3c = 0
14) The system of linear equations x y z 6 ,
is
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
number of solutions, then
a) a = 8, b can be any real number
...
b) Exactly two values of
...
d) Infinitely many values of
...
Let us denote by
a1
a, b, c the determinant a 2
b1
b2
c1
c2
a3
b3
c3
if a, b, c 0 , then the value of x in the
unique
a)
bcd
abc
solution of the above equations is
abd
acd
b) bcd c) d)
abc
abc
abc
18) Consider the system of linear equations
x1 2x 2 x 3 3, 2x1 3x 2 x 3 3,
3x1 5x 2 2x 3 1
...
Suppose
that there are real numbers x, y, z not all zero
such that x = cy + bz, y = az + cx, z = bx + ay
have a
solution, then a 2 b 2 c2 2abc
a) – 1
b) 0
c) 1
d) 2
21) The number of solution of the equations
x 2 x 3 1, x1 2x 3 2, x1 2x 2 3 is
a) Zero
b) One
c) Two
d) Infinite
x 2y 3z 10 and x + 2y + az = b has no
solution when
MHT – CET / JEE (Main)
[ 20 ]
Matrices
Multiple Choice Questions From
MHT CET
a) A + B = B + A and A + (B + C) = (A + B) + C
b) A + B = B + A and AC = BC
c) A + B = B +A and AB = BC
d) AC = BC and A = BC
2 4
3) A
Then A2 =
1 2
0
b)
4
0
d)
0
a) Null matrix
c) Unit matrix
16
0
0
1
a h g
x
4) A = [x y z], B h b f , C y
g f c
z
then ABC =
a) ax by cz 2gx 2fy 2cz
2
2
2
b) ax by cz 2hxy 2by 2cz
2
2
2
c) ax 2 cy 2 bz 2 xy yz zx
d) ax 2 by 2 cz 2 2hxy 2gxz 2fyz
3 3
0
x
5) A 3 0 4 , B y Then B’(AB) is
3 4 0
z
a) Null matrix
b) Unit
c) Singular
d) Symmetric
MHT CET – 2005MHT CET – 2005
1 2
1 3 2
1) If A 3 2 , B
then AB =
4 1 3
1 0
9 1 4
a) 11 7 0
1 3 2
1 4
9
b) 11 7 0
1 3 2
9 1 4
c) 11 7
0
1 3 2
MHT – CET / JEE (Main)
9 1 4
d) 11 7 0
1 3 2
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
MHT CET – 2004
1 2
2
1) If A
Then A – 5A is equal to
3
4
a) 2I
b) 3I
c) –2I d) Null matrix
2 1
1 2
1 3
2) A
,B
, C
then
1 2
2 1
2 1
2)
3)
4)
3 2
AI
then ( A + I ) (A – I ) =
4 1
5
a)
8
1
A
2
4
5 4 5 4
5 4
b)
c)
d)
8 9 8 9
8 9
9
1
1 a
, If (A + B)2 = A2 +
,B
1
4 b
2
B then a and b are
a) 1, –1
b) 1, –2 c) –1, 1
d) 0, 2
2 1 1
If A 2 3 2 then A2 =
4 4 3
a) null matrix
b) it self A
c) unit matrix
d) scalar
MHT CET – 2006MHT CET – 2006
1 0
1 0
1) If A
, B
1 0 , then AB =
0 1
a) a null matrix
b) an identity matrix
c) matrix A
d) matrix B
2)
3)
4)
1)
2)
3)
cos sin
If A( )
, then A 2 ( ) =
sin
cos
a) 1
b) 2
c) 3
d) 0
8 4
5 4
2
, B
If A
, then (A + B) =
10
5
10
8
a) A2 + B2
b) A2 + BA+ B2
c) A2 + AB + B2
d) A2 + 2AB + B2
If A is square matrix of order n, then |KA| =
|A|
a) K | A | b) K n A c) n d) None of
K
these
MHT CET – 2007MHT CET – 2007
2 2
1 1
If A
, B
1 1 , then
2 2
a) A–1 = B
b) B–1 does not exists
c) A–1 does not exists
d) both b & c
5 4
If A
then A–1 =
3 2
a)
1 2 4
2 3 5
b)
c)
1 5 4
2 3 2
1 2 4
d)
2 3 5
1 5 3
2 4 2
Matrix A is of order m x n , matrix B is of order
p x q such that AB exists, then
[ 21 ]
Matrices
a) m = n
4)
c) m = q
1
The matrix A satisfying A
0
3 2
3
a)
b)
6 3
6
d) p = q
5 3 1
is
1 6 0
16
30
3 16
3 3
c)
d)
6 30
6 2
MHT CET – 2008MHT CET – 2008
4 1
2
A
and A 6A 7I 0 , then K =
1
K
a) –2
2)
1)
2)
b) 10
c) –10
d) 2
3 2 6
1 2 2
A 1 1 2 , B 1 3 0 are
2 2 5
0 2 1
a) inverse of each other
b) transpose of each other
c) negative of each other
d) equal
MHT CET – 2009MHT CET – 2009
3 2 4
If A 1 2 1 and Aij are the co factors of aij
3 2 6
then, a11A11 + a12A12 + a13A13 =
a) 8
b) 6
c) 4
d) 0
cos sin
If A
and AB = BA= I then B
sin cos
=
cos
a)
sin
sin
c)
cos
1)
sin
cos
cos sin
b)
sin cos
cos
sin cos
d)
sin
cos sin
MHT CET – 2010MHT CET – 2010
cos sin
Let A
then the inverse of A
sin cos
is
cos sin
a)
sin cos
sin cos
c)
cos sin
2)
cos sin
b)
sin cos
sin cos
d)
cos sin
a b
1
If A
then A is equal to
c
d
MHT – CET / JEE (Main)
1 d b
ad bc c a
c) ad – bc
d) – ad + bc
MHT CET – 2011MHT CET – 2011
cos sin
If A
and AB = BA = I, then
sin cos
the matrix B is
cos sin
cos sin
a)
b)
sin cos
sin cos
a)
1)
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
1)
b) p = n
1
ad bc
cos sin
c)
sin cos
2)
b)
cos sin
d)
sin cos
7 6 1 4 2 3
If A 4 2 3 1 3 0 then A =
1 3 0 7 6 1
1 0 0
a) 0 1 0
0 0 1
1 1 0
b) 0 0 1
0 1 0
0 1 0
0 0 1
c) 0 0 1
d) 0 1 0
1 0 0
1 0 0
MT CET – 2013T CET – 2013
cos sin 0
1) If A sin cos 0 , where A11 ,A12 ,A13
0
0
1
are cofactors of a11 ,a12 ,a13 respectively, then the
value of a11A11 ,a12A12 ,a13A13 is
2)
1)
a) –1
b) 1
c) 0
d) 1/2
1 3 3 x 12
If 1 4 4 y 15 , then the values of x, y,
1 3 4 z 13
z respectively are
a) 1, 2, 3 b) 3, 2, 1
c) 2, 2, 1
d) 1, 1, 2
MHT CET – 2016MHT CET – 2016
1 1 0
If A 2 1 5 , then a11A21 a12A 22 a13A 23
1 2 1
a) 1
2)
[ 22 ]
b) 0
c) –1
d) 2
2 2
0 1
–1 –1 –1
A
,B
then (B A ) =
3
2
1
0
2 2
2 2
2 3
1 1
a)
b) 2 3 c) 2 2 d) 2 3
2
3
Matrices
3)
MHT MHT CET – 2019 ( Online ) 2019 (Online)
1 2
If A
such that AX = I, then X =
4 3
1 1 3
1 4 2
b)
5 2 1
5 4 1
1 3 2
1 1 2
c)
d)
5 4 1
5 1 4
MHT CET – 2017MHT CET – 2017
1 0 0
The inverse of the matrix 3 3 0 is
5 2 1
1)
1
4 3 2
and
A 1
1 2 0
If A
1
a)
2)
3)
3 0 0
1
a)
3 1 0
3
9 2 3
3 0 0
1
b)
3 1 0
3
9 2 3
3 0 0
1
c)
3 1 0
3
9 2 3
3 0 0
1
d) 3 1 0
3
9 2 3
14 1
If the inverse of the matrix 2 3 1 does
6 2 3
not exist then the value of α is
a) 1
b) –1
c) 0
d) –2
10 0
For a invertible matrix A if A adjA
0 10
then A
...
B
...
A then k
k
0 1 1
=
1) 7
2) 11
3) –11
4) –7
1 2 x
If the inverse of matrix A 4 1 7 does
2 4 6
not exist, then x =
...
A
then | A | =
0 10
1) 0
1)
2)
[ 23 ]
2) 10
3) 100
4) 20
MHT CET – 20202020 ( Online )
2 3
1 0
–1
If A
,B
, then (AB) =
1
2
3
1
2 3
2 3
1)
2)
7 11
7 11
2 3
2 3
3)
4)
7 11
7 11
cos sin
–1
If A
, then A =
sin
cos
sin cos
1)
cos sin
sin cos
2)
cos sin
Matrices
cos sin
3)
sin cos
3)
5)
6)
2 1
If A
, such that A2 4A 3I 0,
1 2
then A–1 =
1 2 1
1 2 1
1)
2)
3 1 2
3 1 2
1 2 1
1 2 1
3)
4)
3 1 2
3 1 2
Which of the following matrix is invertible ?
1 2 3
4 2
A1
, A2 4 5 7 ,
2
1
2 4 6
1 0 0
1 0 1
A3 5 2 1 , A 4 0 2 3
7 2 1
1 2 1
1) A4
2) A3
3) A2
4) A1
2 0 0
If A 0 2 0 , then A4 A–1 =
0 0 1
8 0 0
1) 0 8 0
0 0 1
8 0 0
2) 0 8 0
0 0 1
0
0
1/ 2
3) 0 1/ 2 0
0
0
1
4 0 0
4) 0 4 0
0 0 1
2 3
The adjoint of the matrix A
is
3 5
5 3
5 3
1)
2)
3 2
3 2
3)
7)
1 5 3
19 3 2
4)
1 5 3
19 3 2
2 3
2 3
If A
and B
,
1 2
1 2
–1
–1 –1
then (B A ) =
0 1
2 3
1 2
1 0
1)
2)
3)
4)
1 0
1 2
3 4
0 1
8)
1 2 1
1
If A 1 1 1 , then adj adjA
1 1 0
MHT – CET / JEE (Main)
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
4)
1) A–1
cos sin
4)
sin cos
2) I
3) A2
4) 2A
x 2 3
9) The value of x such that the matrix 4 5 6
2 3 5
is not invertible is
10
7
7
10
1)
2)
3)
4)
7
10
10
7
0 0 1
10) If A 0 1 0 , then
1 0 0
1) A–1 = I
2) A is not invertible
–1
3) A = 2A
4) A = A–1
1 0 2
11) If A 2 1 3 , where Aij is the cofactor of
0 3 5
the element aij of matrix A, then
a 21A 21 a 22 A 22 a 23A23
1) 0
2) 26
3) –26
4) –2
12) If the elements of matrix A are the reciprocals of
1 2
elements of matrix 2 1 , where is
2 1
complex cube root of unity, then
1) A–1 = A2
2) A–1 = A
3) A–1 = I
4) A–1 does not exists
1 2
1 2 1
13) If A
, B 2 1 , then (AB)–1 is
2 1 0
0 1
1 5 5
1)
5 4 5
1 5 5
3)
5 4 5
1 0
1
14) If A
,I
1 7
0
then the value of k is
1
1)
2) –7
7
1 5 5
2)
5 4 5
1 5 5
4)
5 4 5
0
and A2 = 8A + kI,
1
3)
1
7
4) 7
1 1 1
1
15) If AX = B, where A 2 1 0 , B 1 ,
3 3 4
2
[ 24 ]
x
and X y , then x + y + z =
z
Matrices
1) 1
1
16) If A
1
1 3
1)
7 4
2) 3
3) 6
4) 2
3) –A
1 3
24) AX = B, where A 1 4
1 3
1
4 1
–1
,B
, then (A + B) =
2
3
1
2
5
3 2
2) 7
4 5
1 3 2
4)
7 4 5
a 1 4
17) The matrix A 3 0 1 is not invertible
1 1 2
only if a =
1) –17
2
18) If A
1
2) –16
3) 16
4) 17
3
1 0
, then B–1 A–1 =
,B
2
3 1
2 3
2 3
1)
2)
7 11
7 11
2 3
2 3
3)
4)
7 11
7 11
19) The sum of the cofactors of the elements of
1 3 2
second row of the matrix 2 0 1 is
5 2 1
b) 3
3) 5
4) –23
3 1 1
2 0 1
1
20) If A 5 1 0
and A 6 5 ,
2 2
0 1 3
then the values of and are respectively
...
1 2
1 2
1 2
1 2
,
,
,
,
1)
2)
3)
4)
11 11
11 11
11 11
11 11
MHT CET – 2021MHT CET - 202
2 2
0 1
1) If A
,
B
1 0 ,
3 2
–1 –1 –1
then (B A ) =
...
a) –1000 b) 100
c) 20
d) –10
1 2 3
13 2 b
If A 1 1 and B 3 1 2
2 4 7
2 0 1
where matrix B is inverse of matrix A, then the
values of a and b are
...
A =
0
0
1
cos sin 0
a) sin cos 0
0
1
0
cos sin 0
b) sin cos 0
0
1
0
cos sin 0
cos sin 0
c) sin
cos 0 d) sin cos 0
0
0
0
1
0
1
5)
6)
7)
0 1 2
If A 1 2 3 , then A–1 =
...
1 1
1
a) ,
b) –1, 1 c) 2,
d) 1, –1
2 2
2
1 0 2
5 x 2
If A 1 1 2 , adj
...
a) 3
b) 6
c) 4
MHT – CET / JEE (Main)
d) 5
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
3)
For a 3 × 3 matrix A,
0
10 0
if A adj
...
If A 1
2 1 2
where I2 is a unit matrix of order 2
...
A) =
, then | A | =
...
A) = AA , then
3
2
5a + b = ?
a) 5
b) 13
c) –1
d) 4
1 2
1 2 1
12) If A
and B 3 1 then
1
1
3
0 2
(AB)–1 =
...
1
1
a)
b) –6
c) 36
d)
36
6
2
k
–1
14) If A
, then A does not exist if k =
2
k
...
Thrice the third
number when added to the first number gives 7
...
The product
of these numbers is
...
a) 4
b) 6
c) 5
d) 3
1 2 3
18) If A 1 1 2 , then A(adj
...
1 2 4
1 1 1
a) 2 1 2
3 2 4
3 0 0
b) 0 3 0
0 0 3
1 2 3
c) 1 1 2
1 2 4
d)
0
0
1/ 3
0
1/ 3
0
0
0
1/ 3
1 2 3
19) If A 1 1 2 and A(adj
...
a) 625
b) 256
c) 81
d) 16
i
1
20) If A
and A does not exist, then
i
λ =
...
1 3 4
a) 13, –6, –5
b) –13, 6, 5
c) –13, –6, 5
d) 13, 5, 6
3 2 6
1
22) If A 1 1 2 , then A =
...
5
2
1
1
a) 19
b) –19
c)
d)
19
19
x1
1 1 1
1
17) If A 2 1 0 , B 1 and X x 2
x 3
3 3 4
2
5 20 2
a) 1 3 0
3 11 1
5 20 2
b) 1 3 0
3 11 1
5 20 2
c) 1 3
0
3 11 1
5 20 2
d) 1 3 0
3 11 1
2 3
1 0
23) If A 1
and B 1
, then
1 2
3 1
AB
1
2 3
a)
7 11
2 7
b)
3 1
2 3
c)
7 11
2 7
d)
3 11
MHT CET – 2022MHT CET - 202
1 3
1) If A a ij 1 2
33
1 1
of aij then a 31 A 31 a 32 A 32
a) 0
b) 20
3
2 and Aij is a cofactor
4
a 33 A 33 is equal to
c) 5
d) 15
2 3
2) If A
, then A + adj A is
4 1
1 3
3 0
1 0
1 3
a)
b)
c)
d)
0 3
0 1
4 2
4 2
3 2 4
3) If A a ij 1 4 1 and Aij is a cofactor of
33
2 6 3
aij, then the value of a 21A 21 a 22A 22 a 23A 23 is
equal to
a) 18
b) 8
c) –8
d) 0
2 3
4) If A
, then adj (3A2 +12A) is equal to
4 1
[ 27 ]
21 63
a)
84 0
21 63
b)
84 0
21 63
c)
0
84
21 63
d)
84 0
Matrices
5) The element in the third row and second column
3 2 6
of the inverse of the matrix 1 1 2 is
2 2 5
b) 1
c) –2
d) 2
1 2 3
6) If A a ij 1 1 5 and Aij is a cofactor of
33
2 4 7
aij, then a11A 21 a12 A 22 a13 A 23 is equal to
a) 1
b) 0
c) 2
d) –1
1 3 2
7) If A 3 0 5 and A(adj A) = KI, then the
2 5 0
value of K is, (where I is unit matrix of order 3)
a) –85
b) 85
c) –25
d) 25
1 2i i 2
0
8) If A 1 2i
0
K and A 1 does not
2 i
7
0
exists, then K =…
...
3 / 2 1/ 2
1 1
If P
, A
and
3 / 2
0 1
1/ 2
Q = PAPT then PTQ2005P is
1
1 2005
1
2)
[ I
...
T
...
I
...
E
...
– 2003]
a) 2ab, a 2 b2
b) a 2 b2 , ab
c) a 2 b2 , 2ab
d) a 2 b 2 , a 2 b2
3)
0
1 0
If A
, A 2 B then value
,B
1 1
5 1
of is
a) 4
4)
b) –1
[ I
...
T
...
I
...
E
...
– 2011]
a) 0
5)
[ 28 ]
b) –H
c) H2
d) H
1 0 0
Let A 2 1 0
...
I
...
E
...
– 2012 ]
a) –2
b) 1
c) 0
d) –1
Let P a ij be a 3 × 3 matrix and let
Q bij , where bij 2i j a ij for 1 i, j 3
...
I
...
–
2012 ]
a) 210
8)
b) 211
c) 212
d) 213
If P is 3×3 matrix such that PT = 2P + I, where
PT is the transpose of P and I is the 3×3 identity
matrix, then there exists a column matrix
x 0
X y 0 such that :
z 0
[ I
...
T
...
I
...
– 2012 ]
a) 4
b) –1
c) 1
d) +2
5 5
10) If A 0 5 , | A2 | = 25, then | α |=
0 0
5
1
a)
5
[ A
...
E
...
E
...
I
...
E
...
– 2012]
1 1 1
4 2 2
11) If A 2 1 3 , 10 B 5 0 k , B
1 1 1
1 2 3
is inverse of A then k =
[A
...
E
...
E
...
I
...
E
...
– 2004 ]
a) A is a zero matrix
b) A = (–1) = I, where I is a unit matrix
c) A 1 does not exists
...
I
...
E
...
– 2005 ]
a) A
b) A + I
c) I – A
d) A – I
1 0
1 0
14) If A
,
I
0 1 , then which of the
1 1
following holds for all n ≥ 1, by the principle of
mathematical induction
...
I
...
E
...
– 2005 ]
a) A 2n1 A n 1 I
b) A nA n 1 I
c) A 2n1 A n 1 I
d) A nA n 1 I
n
n
n
n
1 0 0
1
15) A 0 1 1 and also A 1 A 2 cA dI
6
0 2 4
, where I is a unit matrix, then the ordered pair
(c, d ) is
a) (–6,11 )
[I
...
T
...
I
...
E
...
–2006]
a) A = B
b) AB = BA
c) either A or B is a zero matrix
...
1 2
a 0
17) Let A
and B
, a, b ϵ N then
3 4
0 b
[ 29 ]
Matrices
a) If det A 1, then A 1 need not exist
...
c) If det A 1, then A1 exists and all its
entries are non–integers
...
19) The number of 3×3 non singular matrices, with
four entries as 1 and all other entries as 0 is
[ A
...
E
...
E
...
I
...
– 2010 ]
c) 168
d) 2
JEE Main 2019
cos sin
, then the matrix A 50 When
1) A
sin cos
is equal to
12
1
3
2
1) 2
3
1
2
2
3
2) 2
1
2
3)
1
4) 2
3
2
3
2
1
2
1
2
3
2
MHT – CET / JEE (Main)
1
2
3
2
3
2
1
2
2)
If
e t
A e t
e t
GHONSE MATHS ACADEMY – MHT CET – GHONSE MATHS ACADEMY – MHT CET
[ A
...
E
...
E
...
c) there exists exactly one B such that AB = BA
...
Then which of the following is true?
[ A
...
E
...
E
...
Then
equal to
1) 135
2) 9
3) 10
4) 15
6) If
1 1 1 2 1 3
1 n 1 1 78
,
0 1 0 1 0 1
...
Then, the value of α is
A 32
1 0
32
b) 0
c)
64
d)
16
10) The total number of matrices
0 2y 1
A 2x y 1 , x, y, R, x y for which
2x y 1
A T A 3I3 is
a) 2
b) 4
c) 3
d) 6
JEE Main 2020
1)
Let be a root of equation x 2 x 1 0 and
1 1
1
matrix A
1
3
1 2
1
2 , then the matrix
4
31
2)
A is equal to
1) A3
2) A2
3) I3
4) A
Let A = [aij] and B = [bij] be two 3 × 3 real
matrix
such that bij 3
i j2
a ji ,
where i, j = 1, 2, 3
...
If the matrix A b c a
c a b
satisfies ATA = I, then a value of abc can be
a)
6)
1
3
b)
1
3
c) 3
d)
2
3
x 1
Let A
, x R and A 4 = [aij]
...
a) 10
7)
a ji
MHT – CET / JEE (Main)
31 a12
31 a 22
a11
2 1
81 3 3 3 a 21
[ 31 ]
b) 20
c) 15
d) 25
cos isin
If A
, and
24
i sin cos
a b
A5
, where i 1 , then which one of
c d
the following is not true?
a) a2 – d2 = 0
c) a 2 b 2
1
2
b) a2 – c2 = 1
d) 0 a 2 b 2 1
Matrices
JEE Main 2021
0 2
1) If the matrix A
satisfies ,
K 1
1 0 0
equation A A A 0 4 0 for some
0 0 1
real numbers α and β, then β – α is equal to
...
Then A2025 – A2020 is equal to
1 0 0
a) A6 – A b) A5
c) A5 – A
d) A6
1 0
50
3) If P
, then P is
1
/
2
1
1 0
1 50
a)
b)
25 1
0 1
1 25
1 0
c)
d)
0 1
50 1
1 2 0
2 1 5
4) Let A 2B 6 3 3 and 2A B 2 1 6
...
Then the value of n ∈N for which
5 3
Pn = 5I – 8P is equal to
...
Then the system of
i i
linear equations A8 x 8 has
y 64
a) a unique solution
b) infinitely many solutions
c) no solution
d) exactly two solutions